Maximal Multilinear Commutators on Non-homogeneous Metric Measure Spaces

Abstract

Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T_*$ be the maximal Calderón-Zygmund operator and $\vec{b} := (b_1,\ldots,b_m)$ be a finite family of $\widetilde{\operatorname{RBMO}}(\mu)$ functions. In this paper, the authors establish the boundedness of the maximal multilinear commutator $T_{*,\vec{b}}$ generated by $T_*$ and $\vec{b}$ on the Lebesgue space $L^p(\mu)$ with $p \in (1, \infty)$. For $\vec{b} = (b_1,\ldots,b_m)$ being a finite family of Orlicz type functions, the weak type endpoint estimate for the maximal multilinear commutator $T_{*,\vec{b}}$ generated by $T_*$ and $\vec{b}$ is also presented. The main tool to deal with these estimates is the smoothing technique.